Avogadro's Number

Atoms are quite small. One million atoms lined up side by side would about equal the thickness of a single human hair, about a tenth of a millimeter. Because they are so small, chemists don't count the number of individual atoms, they count the number of moles of atoms.

You have already learned one definition of the mole: a mole is an amount of a substance in grams equal to the formula mass in amu. So a mole of helium (formula mass 4.00 amu) has a mass of 4.00 g; a mole of CO₂ has a mass of 44.01 g, because its formula mass is 44.01 amu.

You will now learn a second, equally correct definition of the mole: a mole of a substance always contains the same number of particles of that substance. So a mole of helium has as many helium atoms as a mole of CO₂ has molecules. This definition can theoretically apply to larger things: you could have a mole of eggs, or indeed a mole of moles (the fuzzy little mammal). However, a mole of moles would be truly gigantic (and would end badly, as the link shows). Moles only make sense when dealing with things as insanely small as atoms and molecules.

Why is this? Well, it's because that "number" I just mentioned - the number of helium atoms in a mole of helium - is big. Really big. Really really really big. It's value is 6.02*10²³, and it is given the name "Avogadro's Number."

Converting With Avogadro's Number

The fact that we have these two separate definitions of the mole is not an accident; the value of Avogadro's number was, in fact, selected so that the two definitions would match. If it had some other value, you would not be able to just say "a mole of something has the same mass in grams as its formula mass in amu," and this feature of the mole is what makes it so useful in chemistry. Where does the number come from? It's actually the conversion factor between grams and amu: 1 g = 6.02*10²³, which makes sense once you know both the definitions of the mole.

You already know that a mole of hydrogen atoms weighs 1.01 grams. Since you now also know that a mole of hydrogen atoms consists of 6.02*10²³ atoms, you can determine the number of atoms in any given mass of hydrogen. You could also determine the mass of any number of hydrogen atoms. The connection between mass and number of atoms (or molecules) is the mole.

Let's see some examples of Avogadro's number in action.

Example 1

How many atoms are there in 0.30 moles of carbon?

The method you learned for solving stoichiometry problems also works for problems that related the mass of a sample to the number of moles it contains or the number of atoms (or molecules) it contains. In doing stoichiometry, we made conversion factors based on the number of grams in a mole, which could be flipped upside-down if needed, depending on the direction of the math.

In this case, we will use a conversion factor equal to the number of atoms (or molecules, or whatever) in a mole, which we know is Avogadro's Number. Here's what that looks like in practice.

We begin by writing down what we are given – the moles of carbon atoms. Next, we place the appropriate units in the conversion factor. In this case, moles go on the bottom so that they will cancel the units moles on the left. Atoms go on top so that the units that remain on the left are the same as the units of the quantity we are looking for on the right. Finally, we insert numbers in the conversion factor that give the relationship between the units that we put there. In this case, 1 mole contains 6.02*10²³ atoms, so the numbers we place in the conversion factor are a 1 on the bottom and Avogadro’s number on the top.

You could also approach this problem using the proportionality/cross-multiplication technique we saw briefly in Lesson 3. Use whatever method is most comfortable for you.

Example 2

What is the mass of 45 million gold atoms?

This question is a little more complicated than the previous one, because it really asks us to do two things: we need to take our number of gold atoms into moles of gold, and apply the molar mass of gold to find the mass of this sample.

One approach is to do these steps sequentially, as shown below. First, we cancel out atoms of gold on the bottom to convert to moles, then we multiply by the molar mass to find the grams of gold in our sample (note that I have converted 45 million into scientific notation).

This is my preferred method of solving this kind of two-part calculation, as it keeps things organized in a way my brain likes. However, you may find it confusing and needlessly long. After all, since we know the two definitions of the mole are equivalent, we know that 6.02*10²³ atoms of gold must have a mass of 197.0 g, meaning we can make a single conversion factor that combines these values. As you can see, this approach leads to the exact same answer.

As always, there a many ways to approach these dimensional analysis problems. Choose whatever technique makes the most sense to you. Take some time now to do the practice problems in your lab workbook dealing with Avogadro's number.